Undergraduate Journal of Mathematical Modeling: One + Two
Volume 4 | 2012 Spring Article 2
2012
The Subset Sum Problem: Reducing Time Complexity of NP- Completeness with Quantum Search
Bo Moon University of South Florida
Advisors: Manoug Manougian, Mathematics and Statistics Jing Wang, Computer Science & Engineering
Problem Suggested By: Jing Wang
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Recommended Citation Moon, Bo (2012) "The Subset Sum Problem: Reducing Time Complexity of NP-Completeness with Quantum Search," Undergraduate Journal of Mathematical Modeling: One + Two: Vol. 4: Iss. 2, Article 2. DOI: http://dx.doi.org/10.5038/2326-3652.4.2.2 Available at: https://scholarcommons.usf.edu/ujmm/vol4/iss2/2 The Subset Sum Problem: Reducing Time Complexity of NP-Completeness with Quantum Search
Abstract The Subset Sum Problem is a member of the NP-complete class, so no known polynomial time algorithm exists for it. Although there are polynomial time approximations and heuristics, these are not always acceptable, yet exact-solution algorithms are unfeasible for large input. Quantum computation offers new insights for not only the Subset Sum Problem but also the entire NP-complete class; most notably, Grover's quantum algorithm for an unstructured database search can be tailored to identify solutions to problems within mathematics and computer science. This paper discusses the physical and conceptual feasibility of quantum computation and demonstrates the utility of quantum search by analyzing the time complexities of the classical dynamic programming algorithm and Grover's algorithm in solving the Subset Sum Problem, evincing the implications this has on the NP-complete class in general.
Keywords NP Completeness, Quantum Search, Subset Sum Problem
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TABLE OF CONTENTS
Problem Statement ...... 3
Motivation ...... 3
Mathematical Description and Solution Approach ...... 4
Discussion ...... 23
Conclusion and Recommendations ...... 25
Nomenclature ...... 26
References ...... 27
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THE SUBSET SUM PROBLEM: REDUCING TIME COMPLEXITY OF NP-COMPLETENESS WITH QUANTUM SEARCH 3
PROBLEM STATEMENT
The Subset Sum Problem is a member of the NP-complete class of computational
problems, having no known polynomial time algorithm. The purpose of this paper is to compare the
time complexities of a quantum search algorithm and a classical dynamic programming algorithm as
solutions to this problem.
MOTIVATION
In computational complexity theory, problems within the NP-complete class have no
known algorithms that run in polynomial time. The study of NP-completeness is significant, as
computer science problems lurk in many guises across a variety of disciplines, from chemical
informatics to networking. As such, identifying a problem as NP-complete will conserve both
time and effort for the computer scientist, who can avoid a fruitless pursuit for an efficient
algorithm by knowing beforehand that there are currently none in existence. Of course, NP-
complete problems can still be solved, but either the input data must be restricted to reasonably
small sizes to accommodate superpolynomial time algorithms or accuracy must be compromised
in implementing faster approximation algorithms, neither of which are amenable conditions.
Although most algorithms for NP-complete problems are just short of a brute-force
search, the immense computational power of a quantum computer may give impetus to more
efficient solutions. A quantum search algorithm provides a
framework for finding a solution in a finite search space defined in terms of the problem, running
faster than a classical algorithm but admittedly still exponential. As such, this paper will
demonstrate the utility of quantum search in solving the NP-comple
algorithm to a specific instance of NP-completeness the Subset Sum Problem. As it shall be
https://scholarcommons.usf.edu/ujmm/vol4/iss2/2 DOI: http://dx.doi.org/10.5038/2326-3652.4.2.2 Moon: The Subset Sum Problem: Reducing Time Complexity of NP-Completene 4 BO MOON
soon revealed, a classical algorithm can do no better than a complete search while a quantum
search runs drastically faster by exploiting the principles of superposition and quantum
parallelism on an unstructured database.
MATHEMATICAL DESCRIPTION AND SOLUTION APPROACH
A. THE SUBSET SUM PROBLEM AND DYNAMIC PROGRAMMING
The Subset Sum Problem is as follows: Given a set of positive integers and a positive
target integer , determine whether there exists a subset of whose elements sum to
(Neapolitan and Naimipour). This problem has been shown to be NP-complete by reduction to
the satisfiability problem, so no known polynomial time algorithm exists (Dasgupta,
Papadimitriou and Vazirani). One possible algorithm is to generate all possible nonempty
subsets of through a depth-first search and to sum the elements in each set in time,
terminating only when is reached. This yields a complete search algorithm with a runtime of
. A slightly faster algorithm in pseudopolynomial time can be achieved with dynamic
programming.
In order to apply dynamic programming, the Subset Sum Problem must exhibit optimal
substructure and overlapping subproblems. Optimal substructure appears when the solution to a
problem relies on the solutions to smaller cases. In the Subset Sum Problem, suppose that one
element of the solution subset is known. The original problem is now reduced to finding a
subset of elements that adds up to , so this subproblem consists of fewer elements
and a smaller sum. Thus, an algorithm for the Subset Sum Problem can utilize optimal
substructure by iterating over all to create the subproblems, iterating over recursively on
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THE SUBSET SUM PROBLEM: REDUCING TIME COMPLEXITY OF NP-COMPLETENESS WITH QUANTUM SEARCH 5
those subproblems until a base case is reached, and then conflating the solutions in order to solve
the original problem, thereby reducing the number of time-consuming operations done.
Overlapping subproblems occur when multiple subproblems consisting of identical parameters
are solved. Consider two subproblems with sets and and corresponding target sums and
such that but . If it is possible to solve either case, then solving the other is
unnecessary, as the succeeding case that gave rise to these subproblems requires only the fact
whether or not it is possible to reach the sum regardless of the set used. As a result, overlapping
subproblems can be avoided by recording solutions to subproblems in a table as they are solved
and referring to it to determine if a particular recursive call is unique or has already been visited,
so fewer cases are solved. Since the Subset Sum Problem possesses both optimal substructure
and overlapping subproblems, a dynamic programming algorithm can be applied.
To exploit optimal substructure, a recursive relationship must be identified such that
solutions to problems of smaller dimensionality are somehow combined. Let the function
return a Boolean value of true if it is possible to add to using the th element in the set and false
otherwise. Given a target sum and an element , the only valid operations are to either use the
element, resulting in a subproblem with target sum and element , or skip the element,
creating another subproblem with target sum and element . Using a logical OR operator on
these two cases in functional form will determine whether it is possible to solve at least one of
them and ultimately the previous recursive call as well. The function returns 1 when ,
indicating that the target sum has been successfully reached, and returns when either is
negative or , which means either the subset sum has surpassed the target or the subset is out
of elements respectively. The subproblems and the base cases can be represented as
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so that will solve the original problem statement.
To avoid overlapping subproblems, the algorithm uses a matrix where the element at the
th row and th column conveniently holds the value . In implementing the function, one
would refer to the matrix prior to evaluating each recursive call in order to check whether the
particular subproblem has already been solved. If so, the element is retrieved, otherwise the
matrix entry is filled in as soon as is resolved so that future subproblems can refer to it.
B. TIME COMPLEXITY OF THE DYNAMIC PROGRAMMING ALGORITHM
Since this dynamic programming algorithm solves each subproblem exactly once and the
space of all subproblems can be represented by an matrix, the time complexity of this
algorithm is equivalent to the total number of elements in the matrix. With rows and
columns, there are subproblems. Therefore, the time and space complexity of this dynamic
programming algorithm is . For a more thorough evaluation of time complexity, let be
the number of distinct sums that must be created with the given set so that the following cases
can explain its pseudopolynomial time behavior.
In the worst case scenario, is greater than or equal to the sum of all elements in the set.
As a result, the algorithm must visit every possible sum in order to determine if can be reached
at all. By elementary number theory, there are distinct nonempty subsets, so is
bounded above as such:
Therefore, the original time complexity (where ) can be rewritten as .
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THE SUBSET SUM PROBLEM: REDUCING TIME COMPLEXITY OF NP-COMPLETENESS WITH QUANTUM SEARCH 7
Consider the alternate case when is less than the sum of all elements in the set. The
algorithm does not consider every possible subset sum since some would clearly be greater than
. Consequently, the algorithm visits only the unique sums under the worst conditions
for this particular case, so . For example, consider the set with .
There are possible subset sums, but only the sums will be considered
because any greater sum is unnecessary, so . Therefore, these conditions preserve
the complexity.
This dynamic programming algorithm is considered to be pseudopolynomial because it
behaves as a polynomial time algorithm for large elements in and relatively small , but it is
not actually polynomial time as previously shown. However, it is reasonable to conclude that its
runtime is because this represents the worst-case conditions according to order of
growth analysis, and one cannot ensure that is indeed bounded by the sum of the elements in
the set. Note that the complete search algorithm given earlier also runs in . Although the
time complexities of both algorithms are identical, the dynamic programming one is generally
faster due to its use of optimal substructure and overlapping subproblems. In fact, this is the
fastest known runtime of any classical algorithm for the Subset Sum Problem.
C. OVERVIEW OF QUANTUM COMPUTATION
A quantum computer is one that manipulates the components of its internal state by
exploiting the principles of quantum mechanics (Mermin). Though both classical and quantum
computers share the elementary functionality of Turing machines, quantum computers utilize
special properties of physical systems that are precisely why quantum algorithms are so efficient
(Rosen). Thus, a basic understanding of quantum computational theory is necessary prior to
implementing a quantum search algorithm.
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In classical computers, all operations and data storage are dependent upon the bit, the
fundamental computational unit. It can assume only one of two discrete states, or , which can
or false, or as binary
numbers when multiple bits are strung together. Analogously, quantum computers rely on
quantum bits, or qubits, for data manipulation. Qubits can also be associated with the
aforementioned states, conventionally written in Dirac notation as and (Viamontes,
Markov and Hayes).
Whereas classical bits can be stored using high and low voltages in computer chips,
qubits must have a representation grounded in the physical world in order to be properly
manipulated. For example, using a hydrogen atom, the states and corresponds to an
electron in the excited state and in the ground state respectively (Dasgupta, Papadimitriou and
Vazirani). Abstractly, however, the linearity of quantum mechanics allows linear algebra to aptly
model qubits as vectors and bit manipulations as unitary transformations.
One of the major differences between qubits and classical bits is that the former are
subject to the superposition principle, which states that a quantum system can exist in a linear
superposition of all possible states (Dasgupta, Papadimitriou and Vazirani). As such, a quantum
state of a single qubit can be represented as a unit vector given by a linear combination of the
set of all states, known as the computational basis (Mermin):
where , the amplitude of a
such that is normalized, i.e. (Mermin)
.
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THE SUBSET SUM PROBLEM: REDUCING TIME COMPLEXITY OF NP-COMPLETENESS WITH QUANTUM SEARCH 9
The quantum state of a multiple qubit system is the tensor product of the individual qubit states;
for instance, given two qubits, the quantum system is characterized by a superposition of four
computational basis states as (Mermin)
where the first number in each bit string corresponds to the first qubit and the second number
corresponds to the second qubit (this bit string is commonly replaced with its decimal
representation). The state vector is also normalized, following the condition (Mermin)
.
In general, an -qubit system yields a computational basis of states, where the sum of the
squared magnitudes of the amplitudes equals one (Mermin):
The superposition principle is one of the factors contributing to the immense
computational power of a quantum computer. First of all, a quantum system is uniquely defined
by the amplitudes of qubits; for moderately large values of , a classical computer would
require an exponentially large amount of memory, but a quantum computer needs only qubits
to store the amplitudes of the computational basis states (Dasgupta, Papadimitriou and Vazirani).
Secondly, since quantum mechanics conforms to linear algebra quite well, one operation on an
-qubit quantum state is equivalent to simultaneous individual operations on the
computational basis. This phenomenon, known as quantum parallelism, is carried out by the
properties of linear transformations and allows for exponentially large number of operations to
be executed instantaneously, another reason why quantum algorithms outperform their classical
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counterparts (Viamontes, Markov and Hayes). Entanglement is another phenomenon that is vital
to many applications of quantum computing, but it will not be discussed here since it holds no
for the Subset Sum Problem are not entangled in a quantum system.
Such an improvement in memory and speed comes with a drawback, however. Although
a qubit is in a linear combination of multiple states, the only way to extract information about
any of them is to make a measurement, the act of observing the state of the qubit (Dasgupta,
Papadimitriou and Vazirani). Unfortunately, merely observing one qubit collapses its mechanical
wave function, forcing its state from the superposition to a single computational
basis state, either or . Similarly, measuring an -qubit system requires the measurement of
each qubit, producing only one state out of a computational basis of
the probability of a particular state arising from a measurement of a system is (Mermin)
.
Note that the normalization condition for the state vector
the magnitude of the vector and the sum of all probabilities is equal to exactly one.
For a given set of qubits, the amplitudes cannot be known, for they are arbitrary complex
numbers; in fact, the system reduces into a single state with an effectively unknown probability,
all amplitudes erased due to measurement. Quantum computers are not deterred, however, as
there remains a way to gather information from qubits.
Since the outcome of a specific state is probabilistic, it is inherent that quantum
algorithms are not deterministic. Classical computers pass bits through electrical circuits and
logic gates in order to manipulate bits, providing the basic functionality algorithms need in order
to operate. Likewise, the general procedure for quantum algorithms is to obtain qubits in a
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THE SUBSET SUM PROBLEM: REDUCING TIME COMPLEXITY OF NP-COMPLETENESS WITH QUANTUM SEARCH 11
known state by measurement and pass them through elementary quantum gates, manipulating the
qubits by enhancing, damping, or inverting amplitudes with the goal of maximizing the
amplitudes and hence the probability of desired states while minimizing those of undesirable
ones (Dasgupta, Papadimitriou and Vazirani). Thus, the goal of the quantum algorithm is to
generate the correct output upon measurement with high probability.
D. QUANTUM GATES
Bit operations in classical computers occur by passing bits through logic gates to be
transformed according to their current states. These gates are physical implementations of the
logical operators AND, OR, XOR, and NOT, and by linking together a series of logic gates, any
computation can theoretically be achieved (Berman, Doolen and Mainieri). Quantum computers
also rely on similar constructions known as quantum gates for bit manipulation. The linearity of
quantum mechanics allows for these quantum gates to be modeled as a linear transformation on
the state vector of the system. The most significant difference between classical logic gates and
quantum ones is the requirement for the matrix representation of any transformation of a
quantum gate to be unitary, that is, the Hermitian conjugate of the matrix is also the inverse
(Nielson and Chuang):
In order for quantum computation to be physically realized, all transformations must
obey the conditions stipulated by the Schrödinger equation (Berman, Doolen and Mainieri).
First, a solution to the Schrödinger equation for a quantum system at an arbitrary moment in time
can be derived from any other known solution (Berman, Doolen and Mainieri). Consequently,
quantum gates must be reversible, as it allows for the qubits to reflect the state of the quantum
computer at any point between computations as long as measurements are not made. Secondly,
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for efficient computation, the Hamiltonian of the system should be time-independent (Berman,
Doolen and Mainieri)
environment upon the erasure of a single bit (Nielson and Chuang). Reversible computation
prevents this flow of energy, preserving the Hamiltonian in turn, because erasing bits is
unnecessary when past states can be reconstructed by undoing operations. Thus, all unitary
matrices satisfy these conditions since they are invertible and norm-preserving. Classical logic
gates fail to meet the criteria, mainly because they are not invertible. For example, the XOR
operation takes as input two bits but outputs only one bit; with a loss of information, it is
impossible to derive the state of each input bit given the state of the output bit.
Reversible counterparts of the logic gates exist for quantum computation, so it is possible
to simulate any classical operation with a quantum computer (Berman, Doolen and Mainieri).
Furthermore, it has been shown that any unitary matrix can be decomposed as (Nielson
and Chuang)
where and are real numbers, allowing for an arbitrary construction for a single qubit
quantum gate provided with the correct parameters. This result applies not only for single qubit
gates, however, as it has been shown that any multiple qubit gate can be constructed using one-
and two-qubit gates (Nielson and Chuang)
not be overly concerned with the physical possibility of quantum gates as long as it is known that
any gate can be created. Although arbitrary gates may be constructed, there are two specific
unitary transform and the Hadamard transform.
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THE SUBSET SUM PROBLEM: REDUCING TIME COMPLEXITY OF NP-COMPLETENESS WITH QUANTUM SEARCH 13
1) The Unitary Transform: The standard unitary transform used by most quantum algorithms is
given by (Mermin)
where represents addition modulo 2 and is some auxiliary function, usually the oracle.
accepts a single vector in the form of a tensor product between and , which are
known as the input and output registers respectively (Mermin). The input register is typically the
state of the quantum system, and the output register is some other n-qubit state that varies with
respect to the specific application of . Furthermore, it is clear that the standard unitary
transform is its own inverse. The most significant feature of is that it allows the output
register to record information about via . In this way, it is possible to gain some
information about without collapsing it by measuring instead, leaving
available for further processing (Mermin).
2) The Hadamard Transform: One of the basic quantum gates, the Hadamard transform acting
on a single qubit yields the following (Dasgupta, Papadimitriou and Vazirani):
Like , is its own inverse. More importantly, the properties of tensor products allows for
multiple Hadamard gates to act on multiple qubits. It can be easily verified that (Mermin)
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otherwise known as the equal superposition of the computational basis. Qubits in the or
state are easily obtainable by measurement, so is extremely useful in generating a uniform
qubit state when qubits only in the state are available.
E. GROVER S ALGORITHM
an unsorted search space with extremely high probability. Since NP-complete problems are
Sum Problem
with several adjustments and a slight computational overhead. For ease of explanation, it will be
assumed for now that there is only one solution to the Subset Sum Problem, as the presence of
additional solutions affects the algorithm only by enhancing the probability of success.
1) The Oracle: The oracle, also known as the black-box function, is a special function whose
implementation is unknown but can be assumed to execute a specific duty reliably in polynomial
time (Viamontes, Markov and Hayes). Oracles are common in quantum algorithms, as they help
to simplify the problem abstraction and divert attention away from the uninteresting
technicalities of constructing such a basic function. For completeness, however, the structure of
the oracle will be discussed.
state is the one sought after. It is important to clarify that this does not imply that the oracle can
find solutions by itself; rather, the oracle can only test possible solutions in polynomial time.
Furthermore, solution states are trapped within the superposition, so the other operations in
oracle.
For the context of the Subset Sum Problem, define a bijection between subsets of
elements and the states of qubits such that the index of a qubit corresponds with the index of an
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THE SUBSET SUM PROBLEM: REDUCING TIME COMPLEXITY OF NP-COMPLETENESS WITH QUANTUM SEARCH 15
element in the set: if the ith qubit of is , include in the subset, but do not use if the
qubit is . Thus, a one-to-one and onto relation is created, so the oracle can be implemented
through this meaningful bijection.
It is necessary to define another reference for the input size as (by
convention, the logarithm is in base 2). That is, represents the smallest power of greater than
. The number of computational basis states must be a power of , so this refinement of is
required in order to record all the correspondences whenever is not already a power of . Of
course, this means that there may be more computational basis states than the bijection should
allow, but this need not be a hindrance because the oracle can identify and skip any invalid state.
The oracle returns 1 if x matches the desired state and returns otherwise:
where is the qubit state whose bit string representation corresponds to a solution subset
according to the bijection. In order to determine whether a given state is indeed the correct bit
string representation, the oracle will follow the bijection, calculate the corresponding subset sum,
and then compare it to the target integer. Since this procedure is essentially the addition of
multiple elements, the time complexity of the oracle is .
By using the standard unitary transformation
the oracle flips the output register if represents the subset of elements whose sum is . By
quantum parallelism, applying one unitary transformation on a state vector is equivalent to
transforming all computational basis states, evidence already that is much
more efficient than the classical algorithm. However, this operation is still insufficient, as
must somehow distinguish the correct state from the incorrect ones, but this unitary
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transformation leaves the input register invariant. One way to mark the desired element is to
apply a phase inversion to the computational basis state by multiplying its amplitude by
while leaving all other states untouched (Morsch). In order to accomplish this, consider the
Hadamard transform on the state ,
When a bit flip is applied,
which is equivalent to multiplying the qubit by . Upon further inspection of the output register
, if the oracle returns a , then the tensor product of the input and output registers
becomes negative, and it is left alone when the oracle returns . Thus, the unitary transform can
be rewritten as (Mermin)
A conceptual implementation of the oracle is now complete. As for its physical implementation,
it has been discussed previously that any quantum gate can be physically realized, so it will not
be of concern here.
2) The Grover Iteration: to be the equal superposition
state obtained by passing n qubits in the state through the Hadamard gate (Mermin):
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THE SUBSET SUM PROBLEM: REDUCING TIME COMPLEXITY OF NP-COMPLETENESS WITH QUANTUM SEARCH 17
Next, the state vector is passed through the standard unitary transformation with in
the input register and in the output register, applying a phase inversion to the amplitudes of
the desired computational basis states:
As discussed earlier, the ancillary qubit is tensored with the input register in order for the
phase inversion to work properly; however, since this particular qubit is not required for any
since the unitary transform indicates a proper tensor product between the input and output
registers, the state vector does not suffer from entanglement, so a measurement can be taken
regardless of the presence of .
Afterwards, a quantum gate is applied, where is the controlled phase
gate (Nielson and Chuang). The action of is to invert the phase of every computational basis
state save for , which can be described by (Nielson and Chuang). The net
effect of applying Hadamard transforms before and after the controlled phase gate allows for
to be rewritten as the diffusion matrix (Nielson and Chuang)
where is the equal superposition state that remains invariant within the gate. Essentially, the
diffusion matrix represents the inversion about the mean operation (Grover). For any given
amplitude of a computational basis state and the mean of all amplitudes , the amplitude is
below the mean by exactly . The diffusion matrix replaces the amplitude with a new
quantity that is above the mean by the same amount that was below the mean, or
thereby inverting the amplitude of the state about the mean (Morsch).
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The inversion about the mean operation is essential for the magnification of the desired
amplitude and hence the probability of correct output. Since the oracle function inverted the
phase of the desired state, it is farther below the mean amplitude compared to the other
amplitudes. After applying the diffusion matrix, the desired amplitude will be farther above the
mean than the other amplitudes, which will have decreased. For example, in Figure 1, the
amplitudes of four different states are represented by vertical bars, and C is the farthest away
from the average line due to its inverted phase. However, after the inversion about the mean
operation, the amplitudes are above the mean by exactly the same amount they were once below
it; since C was the farthest below, it now becomes the farthest above.
Figure 1: The inversion about the mean operation on the amplitudes of four states (Grover).
The oracle function and the inversion about the mean operation together constitute the
Grover iteration, the part of the algorithm that must be repeated several times in order to improve
probability of success. As such, the Grover iteration is applied repeatedly on the initial
input (Mermin). Figure 2 demonstrates the results of the first Grover iteration on
the amplitudes of the computational basis states. For up to a certain number of iterations, the
amplitude of the desired state will continuously increase while the others approach . At this
point, a measurement is carried out to obtain one possible solution state , and due to repeated
use of G, the probability of observing a desired computational basis state is close to 1.
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THE SUBSET SUM PROBLEM: REDUCING TIME COMPLEXITY OF NP-COMPLETENESS WITH QUANTUM SEARCH 19
Figure 2: Single call of the Grover iteration on the equal superposition state (Grover).
Since the Subset Sum Problem is not concerned with what particular subset is the
solution, for there could be multiple subsets, the final unitary transform is applied:
Simply measuring the output register will reveal whether is the correct solution: if the
measurement produces , then the state is correct, or else indicates a faulty solution.
3) Geometric Interpretation:
transformations acting upon vectors in space, providing a medium for deriving the time
complexity of this quantum algorithm. First of all, one must realize that the computational basis
is in fact an orthonormal basis of a Hilbert space, as any arbitrary state vector within the vector
space may be created with complex linear combinations of the computational basis states. Given
a computational basis of states, let there be possible solutions to the Subset Sum Problem.
Furthermore, let be the set of all computational basis states and B be the subset representing all
desirable states. Two new vectors can be created with linear combinations of the vectors in and
its complement (Nielson and Chuang):
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In other words, is the linear combination of all undesirable states and is the linear
combination of all desirable ones, both of which have been normalized. Therefore, the initial
input state of equal superposition
can be rewritten as a linear combination of and , normalizing once more (Nielson and
Chuang):
Note that , and reside within the same vector space, as they are normalized and are
linear combinations of the basis vectors.
The computational basis states are orthonormal, as each is normalized and any pair of
vectors is orthogonal. Since and are linear combinations from disjoint sets of an
orthonormal basis, they are orthonormal as well. Consequently, and together span a
plane, forming the basis of a two dimensional Hilbert space. The most important corollary to this
realization is that state vectors may be rewritten as linear combinations of arbitrarily defined
computational basis states; that is, through a change of basis, qubits may be subjected to
measurement with respect to an entirely new yet equivalent basis (Nielson and Chuang). As a
result, it is fitting to describe the vectors , and as equivalent to
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THE SUBSET SUM PROBLEM: REDUCING TIME COMPLEXITY OF NP-COMPLETENESS WITH QUANTUM SEARCH 21
where the column vectors for and are defined by Dirac notation. Therefore, the entire
quantum system can be considered to take place within a plane (Nielson and Chuang).
and are very nearly orthogonal, but and are entirely orthogonal, so
and are separated by a very small angle , which can be found by taking the inner product
(Mermin)
where the last approximation is highly accurate for large N by the Small Angle Approximation.
As shown in Figure 3,
result of which is a rotation. First, the unitary transformation with the oracle function reflects the
initial vector across because the phase inversion on reverses the direction of the
corresponding component in :
After , will be below , displaced a total of . Similarly, the inversion about the
mean operation reflects across , which happens to be equivalent to
in the first Grover iteration (Mermin). brings towards by , creating a net
rotation of radians for a new state . Therefore, each application of the Grover
iteration rotates the state vector towards , the superposition of all desirable states, by
(Nielson and Chuang). After enough iterations, the final state vector is extremely close to , so
the amplitude of is drastically decreased while the amplitude of experiences a
https://scholarcommons.usf.edu/ujmm/vol4/iss2/2 DOI: http://dx.doi.org/10.5038/2326-3652.4.2.2 Moon: The Subset Sum Problem: Reducing Time Complexity of NP-Completene 22 BO MOON
corresponding increase. By is close to one, and by
the original definition of being the linear combination of all desirable states, any one of those
is equally likely to appear.
Figure 3: Rotation of state vector in a two dimensional Hilbert space (not drawn to scale) (Mermin).
4) Time Complexity of the Quantum Algorithm:
until it is almost nearly to maximize the probability that one of the desirable states will
be selected, so an optimal number of iterations must be calculated. As mentioned before,
and are almost orthogonal, so the state vector must be rotated by approximately radians.
Since the Grover iteration rotates by radians each time, the quantum computer must apply
about times (Vazirani). In order to attain proper time complexity by order of growth
analysis, this quantity must involve the input size. From earlier,
so it follows that
or for a worst-bound case when ,
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THE SUBSET SUM PROBLEM: REDUCING TIME COMPLEXITY OF NP-COMPLETENESS WITH QUANTUM SEARCH 23
(Grover).
DISCUSSION
Since classical and quantum computation differ fundamentally by the type of operations
carried out, the time complexity of quantum algorithms is conventionally written in terms of the
number of oracle calls rather than operations. The time complexity of the dynamic programming
algorithm given previously will also be converted in order to give an accurate comparison of
efficiency.
The dynamic programming algorithm does not use oracle calls, but it is still possible to
rewrite its order of growth with respect to the time complexity of the oracle. As discussed, the
oracle essentially sums elements in the list according to the bit string passed to it, so its
performance is . The number of linear time operations used by the dynamic programming
algorithm is , so oracle calls are required. However, uses
oracle calls, so it significantly outperforms the classical approach by a factor of , a
quadratic reduction in time.
algorithm: the runtime of a quantum search is the square root of the runtime of the analogous
classical search (Nielson and Chuang).
One of the most promising applications of quantum search is to resolve the exponential
time complexity that is characteristic of NP-
provides a quadratic improvement in speed, one may conjecture that an even faster quantum
algorithm, perhaps
https://scholarcommons.usf.edu/ujmm/vol4/iss2/2 DOI: http://dx.doi.org/10.5038/2326-3652.4.2.2 Moon: The Subset Sum Problem: Reducing Time Complexity of NP-Completene 24 BO MOON
algorithm is optimal, so the fastest time complexity of any search algorithm, classical or
quantum, is (Zalka). Moreover, the NP-complete class is defined to be a set of search
problems that reduce to one another, yet the fastest possible search algorithm is still exponential,
which severely diminishes the hope that NP-complete problems do have a polynomial time
solution. However, there exist several alternatives, though not promising, to consider.
It is widely believed that NP-complete problems are intractable because there is no
exploitable organization in the underlying structure of the search space (Nielson and Chuang). In
fact, it is precisely this property that makes certain classical deterministic algorithms so efficient;
polynomial time since the solution satisfies the greedy property of optimizing locally to optimize
globally. As such, it may be possible to expedite solutions to the NP-complete class by
reexamining the search space in novel ways. For example, the dynamic programming algorithm
for the Subset Sum Problem presented earlier runs in pseudopolynomial time by utilizing optimal
substructure and overlapping subproblems. Though this approach was not sufficiently fast, it
demonstrates that there could be some property of the Subset Sum Problem, and possibly the
entire NP-complete class, not yet observed that can be used as the basis of an even faster
algorithm.
Another possibility to consider is the application of reductions. Finding the prime
factorization of an integer is a famously hard problem (though not NP-complete) that has no
known polynomial time solution in the classical realm, but quantum computation offers one.
problem of identifying periodicity, which can be solved with the Fourier transform and some
number theoretic insights (Morsch). Similarly, reductions may help bring known polynomial
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THE SUBSET SUM PROBLEM: REDUCING TIME COMPLEXITY OF NP-COMPLETENESS WITH QUANTUM SEARCH 25
time solutions to the NP-complete problems. Of course, the largest and most overwhelming
obstacle to this consideration by far is that for an NP-complete problem to reduce to P, it would
require the proof of , a puzzle that has eluded generations of computer scientists. Still,
much of computational complexity theory analyzes problems in respect to classical algorithms,
so perhaps reductions of the NP-complete class may be possible with advances in quantum
computation.
Since the search for a polynomial time algorithm for NP-complete problems is bleak,
other approaches must be used for the time b
algorithm is optimal and in fact has just been shown to solve the Subset Sum Problem, and since
all NP-complete problems reduce to one another, quantum search certainly works on the entire
complexity class. Therefore, it is not only feasible but also more efficient to apply quantum
search techniques on NP-complete problems, especially when exact solutions are required, rather
than relying on heuristics and approximation algorithms.
CONCLUSION AND RECOMMENDATIONS
The Subset Sum Problem remains part of the NP-
achieves a notable improvement in time complexity over any known classical algorithm.
Furthermore, by reductions and slight computational overhead, quantum search can be applied to
any instance of NP-completeness, evincing the possibility of diverse application across fields as
well, such as in primality testing or cryptography. Although the physical aspects of quantum
computers were not discussed in depth, they do present significant but definitely tractable
challenges for engineers. Nevertheless, it is exciting to consider that quantum computational
theory may yield in the near future new algorithms inconceivable in a classical domain yet
innovative and possibly revolutionary for all of computer science.
https://scholarcommons.usf.edu/ujmm/vol4/iss2/2 DOI: http://dx.doi.org/10.5038/2326-3652.4.2.2 Moon: The Subset Sum Problem: Reducing Time Complexity of NP-Completene 26 BO MOON
NOMENCLATURE
Symbol Description Big-O notation. For an algorithm with input data size and processing time , indicates that is bounded above by .
Ket; in Dirac notation, a column vector. If is a nonnegative integer, its binary form represents a tensor product of a sequence of and kets.
Vector representing the state of a quantum system; i.e. a complex linear combination of possible discrete states the unmeasured quantum system can assume. Defined in Dirac notation as the column vectors and respectively.
Bra; or a row vector.
Inner product between and .
Outer product between and .
Tensor product between and .
-fold tensor product of with itself.
XOR operator; i.e. bitwise addition modulo 2.
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THE SUBSET SUM PROBLEM: REDUCING TIME COMPLEXITY OF NP-COMPLETENESS WITH QUANTUM SEARCH 27
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